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A166597
Let p = largest prime <= n, with p(0)=p(1)=0, and let q = smallest prime > n; then a(n) = q-p.
2
2, 2, 1, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 2, 2, 6, 6, 6, 6, 6, 6, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 2, 2, 4, 4
OFFSET
0,1
COMMENTS
Note the large prime gap of 72 between 31397 and 31469. This is the prime gap with the largest merit (cf. A111870), 72/log(31397)=6.95352 for primes less than 100000. Also 72/(log(31397))^2=0.67154 (cf. conjectures of Cramer-Granville, Shanks and Wolf) is largest for primes less than 100000. - Daniel Forgues, Oct 23 2009
LINKS
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture.
EXAMPLE
a(0) = 2 since the least prime greater than 0 is 2 (gap of 2 from 0 to 2).
a(9) = 4 since the least prime greater than 9 is 11 (gap of 4 from 7 to 11).
a(11) = 2 since the least prime greater than 11 is 13 (gap of 2 from 11 to 13).
MATHEMATICA
f[n_]:=Module[{a=If[PrimeQ[n], n, NextPrime[n, -1]]}, NextPrime[n]-a]; Join[{2, 2}, Array[f, 120, 2]] (* Harvey P. Dale, May 17 2011 *)
PROG
(PARI) a(n) = nextprime(n+1) - precprime(n); \\ Michel Marcus, Mar 02 2023
CROSSREFS
Cf. A111870. - Daniel Forgues, Oct 23 2009
See A327441 for the classic G(n) version. - N. J. A. Sloane, Sep 11 2019
Sequence in context: A339383 A198260 A029350 * A375325 A302490 A316841
KEYWORD
nonn
AUTHOR
Daniel Forgues, Oct 17 2009
EXTENSIONS
Definition rephrased by N. J. A. Sloane, Oct 25 2009
STATUS
approved